Moises Caicedo's blistering second-half strike cancelled out a Bruno Fernandes penalty as Chelsea salvaged a 1-1 draw with Manchester United on Sunday.
Leandro Trossard transformed the mood at Arsenal with the winning goal as they recovered from an early setback to beat Tottenham Hotspur 2-1 at home.
Images from Sunday's action in the Premier League.
Manchester City have now won only once in 11 games in all competitions.
A summary of Thursday's action in the Premier League.
Manchester City thrashed West Ham United 4-1 in the Premier League on Saturday to claim their first back-to-back wins since October.
IMAGES from the English Premier League matches played on Sunday
IMAGES from the Premier League matches played across England, on Sunday.
Tottenham Hotspur have appointed former Celtic coach Ange Postecoglou as their new manager on a four-year contract.
Championship (second tier) side Rotherham, who haven't made it through to the fourth round of the competition since 2001-02, kept their composure early on as Fulham controlled the play.
A summary of Sunday's action in the Premier League.
Manchester City clinched a 1-0 victory at Tottenham Hotspur, while Chelsea were held to a 0-0 draw at home by Aston Villa.
IMAGES from the Premier League matches played on Saturday.
IMAGES from the Premier League matches played on Saturday
Tottenham Hotspur opened a five-point lead at the top of the Premier League with a 2-1 victory at Crystal Palace as their unbeaten start to the season continued on Friday.
Australia is weighing up a bid to host the 2023 Asian Cup following China's decision to step away as original hosts
Manchester United's Alejandro Garnacho scored with a magical overhead kick in the third minute to send the visitors on their way to a 3-0 victory over an Everton team already reeling in their first Premier League game after being docked 10 points.
Mohamed Salah scored a late equaliser from a penalty kick as Liverpool escaped Old Trafford with a single point in a 2-2 draw with Manchester United to climb to even on points with Premier League leaders Arsenal.
Tottenham Hotspur beat Nottingham Forest 2-0 away with goals by Richarlison and Dejan Kulusevski for a second straight win that nudged them closer to the Premier League top four on Friday although they had midfielder Yves Bissouma sent off late on.
Tottenham Hotspur hammered 10-man Aston Villa 4-0 in the Premier League on Sunday thanks to four second-half goals, allowing them to close the gap with their Birmingham rivals to two points in the chase for Champions League football next season.
Arsenal had more than twice as many chances but failed to take any as the visitors defended gamely.
Australia kick off their World Cup on Friday against Chile, tournament dark horses that new coach Ange Postecoglou actually wants to emulate in his reinvention of the Socceroo squad.
Bernardo Silva and Jack Grealish scored three minutes apart as Manchester City came from behind to beat Premier League newcomers Luton Town.
IMAGES from the English Premier League matches played on Sunday
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Wolverhampton Wanderers stunned Tottenham Hotspur 2-1 with stoppage-time goals from Pablo Sarabia and Mario Lemina on Saturday, thwarting the visitors in their bid to reclaim top spot in the Premier League table.
Australia forward Tim Cahill says they can draw positives from their opening 3-1 loss to Chile at the World Cup on Friday but along with coach Ange Postecoglou he regrets showing them too much respect early on.
Liverpool snatched three points with a wild 1-0 victory over relegation-threatened Nottingham Forest.
'This modern-day recovery, matchups, data, sports science, I'm not sure. I would've said just let me be.'
Bukayo Saka scored either side of halftime as Arsenal crushed second-bottom Burnley 5-0.
Images from the English Premier League matches played over the weekend.
Images from the English Premier League matches played on Sunday night.
Champions Manchester City slipped to a shock first Premier League defeat of the season at Wolverhampton Wanderers on Saturday, while Manchester United's campaign of woe continued with a loss to Crystal Palace.
Images from the English Premier League matches played on Saturday.
IMAGES from the Premier League matches played on Saturday
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Images from the English Premier League matches played on Sunday
Images from the English Premier League matches played on Saturday night.
IMAGES from the English Premier League matches played on Saturday.
Qualifiers Spain and the United States suffered blows in friendlies, the world champions beaten 1-0 by eliminated South Africa on the SoccerCity pitch where they clinched their 2010 crown and the Americans losing 1-0 away to lowly Austria.
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